Using a Put Delta to Size an Equity Index Hedge
Summary
The document presents an exam-style question about protecting a diversified portfolio that tracks the S&P 500 against a loss beyond a specified threshold. It describes a put-based hedge and an alternative replication using European calls, then asks how much of the portfolio should initially be sold and placed in risk-free securities. The poster proposes investing the present value of the protected threshold amount, but is puzzled by an answer key that uses a put option’s delta to determine the allocation.
The example highlights that the two approaches are not interchangeable calculations: the call-based replication addresses the option payoff at maturity, while the stated answer uses the put’s current sensitivity to the index to estimate the stock exposure in a replicating position. The document reports the given delta and resulting allocation, but does not include a full derivation. Its figures depend on the hypothetical assumptions provided, including the option model inputs, and should not be treated as a general hedge ratio.
Key ideas
- The problem concerns downside protection for an index-tracking equity portfolio.
- A proposed call-based replication combines calls with risk-free lending.
- The cited solution uses put delta to estimate the initial allocation to risk-free securities.
- The document gives an answer-key figure but does not derive it in detail.
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Full text
# Basic Replication Strategies
# Basic Replication Strategies
I am studying for a quantitative finance exam with the Society of Actuaries. I am reviewing a previously published exam and its solutions (below). I am wondering if someone could help me understand part (c) as follows:
> You are managing a well-diversified portfolio that mirrors the performance of S&P 500. For simplification, the following assumptions are made: • The portfolio value is $500 million USD. • The price of S&P 500 is 5,000. • The risk-free interest rate is 4% per annum, continuously compounding. • The dividend yield on both the portfolio and the S&P 500 is 0%. • The volatility of the S&P 500 index is 20% per annum. • You can buy/sell S&P 500 underlying stocks to replicate the index movement. You seek downside protection against a decline of more than 10% in the value of the portfolio over the next year. (b) Describe an alternative strategy using traded European call options to achieve the same protection. (c) Calculate the portion of the portfolio to be sold initially and invested in risk-free securities to achieve the same protection.
To demonstrate my current work and understanding, Part (a) had to do with using European Puts to buy the protection, and I got this part right. For part (b), I think I have the right solution as follows:
- Sell current portfolio off
- Buy 500M/5K = 100,000 Call Options with strike K=4,500
- Lend 450M * $e^{-0.04*1}$ at the risk-free rate.
However, Part (c) is where I am extremely confused. My answer was as follows:
- Risk Free Investment = 450M * $e^{-0.04}$ = 432.4M = 86.47% of the portfolio value.
But the answer key says the following SoA Question 5 part c:
> The delta of one put option is $e^{-qT}[N(d_1)-1]=-0.2042$. This shows that 20.42% of the portfolio, about 102.1M, should be invested in risk-free securities
Could someone explain what they mean by this and what is wrong with my attempted solution?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.